What is Google Colaboratory?
Colaboratory, or "Colab" for short, allows you to write and execute Python in your browser, with
- Zero configuration required
- Free access to GPUs
- Easy sharing
Whether you're a student, a data scientist or an AI researcher, Colab can make your work easier. Watch Introduction to Colab to learn more, or just get started below!
import nbconvert
nbconvert
!jupyter nbconvert --to html HostedTemplateRecSysAug17.ipynb
Hit the left-side â–º Play Button to run the code (Double-click this text to see all imported modules, and double-click again to hide)
# @title Imports (Double-click this text to see all imported modules, and double-click this text again again to hide)
from __future__ import print_function
print("loading modules and packages from web")
import numpy as np
import pandas as pd
import collections
from mpl_toolkits.mplot3d import Axes3D
from IPython import display
from matplotlib import pyplot as plt
import sklearn
import sklearn.manifold
import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()
tf.logging.set_verbosity(tf.logging.ERROR)
# Add some convenience functions to Pandas DataFrame.
pd.options.display.max_rows = 10
pd.options.display.float_format = '{:.3f}'.format
def mask(df, key, function):
"""Returns a filtered dataframe, by applying function to key"""
return df[function(df[key])]
def flatten_cols(df):
df.columns = [' '.join(col).strip() for col in df.columns.values]
return df
pd.DataFrame.mask = mask
pd.DataFrame.flatten_cols = flatten_cols
# Install Altair and activate its colab renderer.
print("Installing Altair...")
!pip install git+git://github.com/altair-viz/altair.git
import altair as alt
alt.data_transformers.enable('default', max_rows=None)
alt.renderers.enable('colab')
print("Done installing Altair.")
# Install google spreadsheets and import authentication module.
USER_RATINGS = False
!pip install --upgrade -q gspread
from google.colab import auth
import gspread
from oauth2client.client import GoogleCredentials
We then download the MovieLens Data, and create DataFrames containing movies, users, and ratings.
# @title Load the MovieLens data (run this cell).
# Download MovieLens data.
print("Downloading movielens data...")
from urllib.request import urlretrieve
import zipfile
urlretrieve("http://files.grouplens.org/datasets/movielens/ml-100k.zip", "movielens.zip")
zip_ref = zipfile.ZipFile('movielens.zip', "r")
zip_ref.extractall()
print("Done. Dataset contains:")
print(zip_ref.read('ml-100k/u.info'))
# Load each data set (users, movies, and ratings).
users_cols = ['user_id', 'age', 'sex', 'occupation', 'zip_code']
users = pd.read_csv(
'ml-100k/u.user', sep='|', names=users_cols, encoding='latin-1')
ratings_cols = ['user_id', 'movie_id', 'rating', 'unix_timestamp']
ratings = pd.read_csv(
'ml-100k/u.data', sep='\t', names=ratings_cols, encoding='latin-1')
# The movies file contains a binary feature for each genre.
genre_cols = [
"genre_unknown", "Action", "Adventure", "Animation", "Children", "Comedy",
"Crime", "Documentary", "Drama", "Fantasy", "Film-Noir", "Horror",
"Musical", "Mystery", "Romance", "Sci-Fi", "Thriller", "War", "Western"
]
movies_cols = [
'movie_id', 'title', 'release_date', "video_release_date", "imdb_url"
] + genre_cols
movies = pd.read_csv(
'ml-100k/u.item', sep='|', names=movies_cols, encoding='latin-1')
# Since the ids start at 1, we shift them to start at 0.
users["user_id"] = users["user_id"].apply(lambda x: str(x-1))
movies["movie_id"] = movies["movie_id"].apply(lambda x: str(x-1))
movies["year"] = movies['release_date'].apply(lambda x: str(x).split('-')[-1])
ratings["movie_id"] = ratings["movie_id"].apply(lambda x: str(x-1))
ratings["user_id"] = ratings["user_id"].apply(lambda x: str(x-1))
ratings["rating"] = ratings["rating"].apply(lambda x: float(x))
# Compute the number of movies to which a genre is assigned.
genre_occurences = movies[genre_cols].sum().to_dict()
# Since some movies can belong to more than one genre, we create different
# 'genre' columns as follows:
# - all_genres: all the active genres of the movie.
# - genre: randomly sampled from the active genres.
def mark_genres(movies, genres):
def get_random_genre(gs):
active = [genre for genre, g in zip(genres, gs) if g==1]
if len(active) == 0:
return 'Other'
return np.random.choice(active)
def get_all_genres(gs):
active = [genre for genre, g in zip(genres, gs) if g==1]
if len(active) == 0:
return 'Other'
return '-'.join(active)
movies['genre'] = [
get_random_genre(gs) for gs in zip(*[movies[genre] for genre in genres])]
movies['all_genres'] = [
get_all_genres(gs) for gs in zip(*[movies[genre] for genre in genres])]
mark_genres(movies, genre_cols)
# Create one merged DataFrame containing all the movielens data.
movielens = ratings.merge(movies, on='movie_id').merge(users, on='user_id')
# Utility to split the data into training and test sets.
def split_dataframe(df, holdout_fraction=0.1):
"""Splits a DataFrame into training and test sets.
Args:
df: a dataframe.
holdout_fraction: fraction of dataframe rows to use in the test set.
Returns:
train: dataframe for training
test: dataframe for testing
"""
test = df.sample(frac=holdout_fraction, replace=False)
train = df[~df.index.isin(test.index)]
return train, test
I. Exploring the Movielens Data¶
Before we dive into model building, let's inspect our MovieLens dataset. It is usually helpful to understand the statistics of the dataset.
movielens.head()
Users
We start by printing some basic statistics describing the numeric user features.
users.head()
users.describe()
We can also print some basic statistics describing the categorical user features
users.describe(include=[np.object])
We can also create histograms to further understand the distribution of the users. We use Altair to create an interactive chart.
# @title Altair visualization code (run this cell)
# The following functions are used to generate interactive Altair charts.
# We will display histograms of the data, sliced by a given attribute.
# Create filters to be used to slice the data.
occupation_filter = alt.selection_multi(fields=["occupation"])
occupation_chart = alt.Chart().mark_bar().encode(
x="count()",
y=alt.Y("occupation:N"),
color=alt.condition(
occupation_filter,
alt.Color("occupation:N", scale=alt.Scale(scheme='category20')),
alt.value("lightgray")),
).properties(width=300, height=300, selection=occupation_filter)
# A function that generates a histogram of filtered data.
def filtered_hist(field, label, filter):
"""Creates a layered chart of histograms.
The first layer (light gray) contains the histogram of the full data, and the
second contains the histogram of the filtered data.
Args:
field: the field for which to generate the histogram.
label: String label of the histogram.
filter: an alt.Selection object to be used to filter the data.
"""
base = alt.Chart().mark_bar().encode(
x=alt.X(field, bin=alt.Bin(maxbins=10), title=label),
y="count()",
).properties(
width=300,
)
return alt.layer(
base.transform_filter(filter),
base.encode(color=alt.value('lightgray'), opacity=alt.value(.7)),
).resolve_scale(y='independent')
Next, we look at the distribution of ratings per user. Clicking on an occupation in the right chart will filter the data by that occupation. The corresponding histogram is shown in blue, and superimposed with the histogram for the whole data (in light gray). You can use SHIFT+click to select multiple subsets.
What do you observe, and how might this affect the recommendations?
#@title Distribution of User Ratings
users_ratings = (
ratings
.groupby('user_id', as_index=False)
.agg({'rating': ['count', 'mean']})
.flatten_cols()
.merge(users, on='user_id')
)
# Create a chart for the count, and one for the mean.
alt.hconcat(
filtered_hist('rating count', '# ratings / user', occupation_filter),
filtered_hist('rating mean', 'mean user rating', occupation_filter),
occupation_chart,
data=users_ratings)
Movies¶
It is also useful to look at information about the movies and their ratings.
#@title Movie filter and load
movies_ratings = movies.merge(
ratings
.groupby('movie_id', as_index=False)
.agg({'rating': ['count', 'mean']})
.flatten_cols(),
on='movie_id')
genre_filter = alt.selection_multi(fields=['genre'])
genre_chart = alt.Chart().mark_bar().encode(
x="count()",
y=alt.Y('genre'),
color=alt.condition(
genre_filter,
alt.Color("genre:N"),
alt.value('lightgray'))
).properties(height=300, selection=genre_filter)
(movies_ratings[['title', 'rating count', 'rating mean']]
.sort_values('rating count', ascending=False)
.head(10))
(movies_ratings[['title', 'rating count', 'rating mean']]
.mask('rating count', lambda x: x > 20)
.sort_values('rating mean', ascending=False)
.head(10))
# @title Distribution of movie ratings and average rating per movie.
alt.hconcat(
filtered_hist('rating count', '# ratings / movie', genre_filter),
filtered_hist('rating mean', 'mean movie rating', genre_filter),
genre_chart,
data=movies_ratings)
Exercise 1: Build a tf.SparseTensor representation of the Rating Matrix.¶
In this exercise, we'll write a function that maps from our ratings DataFrame to a tf.SparseTensor.
Hint: you can select the values of a given column of a Dataframe df using df['column_name'].values.
#@title Build a tf.SparseTensor representation of the Rating Matrix
def build_rating_sparse_tensor(ratings_df):
"""
Args:
ratings_df: a pd.DataFrame with `user_id`, `movie_id` and `rating` columns.
Returns:
a tf.SparseTensor representing the ratings matrix.
"""
indices = ratings_df[['user_id', 'movie_id']].values
values = ratings_df['rating'].values
return tf.SparseTensor(
indices=indices,
values=values,
dense_shape=[users.shape[0], movies.shape[0]])
def sparse_mean_square_error(sparse_ratings, user_embeddings, movie_embeddings):
"""
Args:
sparse_ratings: A SparseTensor rating matrix, of dense_shape [N, M]
user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
dimension, such that U_i is the embedding of user i.
movie_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
dimension, such that V_j is the embedding of movie j.
Returns:
A scalar Tensor representing the MSE between the true ratings and the
model's predictions.
"""
predictions = tf.gather_nd(
tf.matmul(user_embeddings, movie_embeddings, transpose_b=True),
sparse_ratings.indices)
loss = tf.losses.mean_squared_error(sparse_ratings.values, predictions)
return loss
def sparse_mean_square_error(sparse_ratings, user_embeddings, movie_embeddings):
"""
Args:
sparse_ratings: A SparseTensor rating matrix, of dense_shape [N, M]
user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
dimension, such that U_i is the embedding of user i.
movie_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
dimension, such that V_j is the embedding of movie j.
Returns:
A scalar Tensor representing the MSE between the true ratings and the
model's predictions.
"""
predictions = tf.reduce_sum(
tf.gather(user_embeddings, sparse_ratings.indices[:, 0]) *
tf.gather(movie_embeddings, sparse_ratings.indices[:, 1]),
axis=1)
loss = tf.losses.mean_squared_error(sparse_ratings.values, predictions)
return loss
#@title Solution # mark for deletion
def sparse_mean_square_error(sparse_ratings, user_embeddings, movie_embeddings):
"""
Args:
sparse_ratings: A SparseTensor rating matrix, of dense_shape [N, M]
user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
dimension, such that U_i is the embedding of user i.
movie_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
dimension, such that V_j is the embedding of movie j.
Returns:
A scalar Tensor representing the MSE between the true ratings and the
model's predictions.
"""
predictions = tf.gather_nd(
tf.matmul(user_embeddings, movie_embeddings, transpose_b=True),
sparse_ratings.indices)
loss = tf.losses.mean_squared_error(sparse_ratings.values, predictions)
return loss
#@title Alternate Solution # mark for deletion
def sparse_mean_square_error(sparse_ratings, user_embeddings, movie_embeddings):
"""
Args:
sparse_ratings: A SparseTensor rating matrix, of dense_shape [N, M]
user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
dimension, such that U_i is the embedding of user i.
movie_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
dimension, such that V_j is the embedding of movie j.
Returns:
A scalar Tensor representing the MSE between the true ratings and the
model's predictions.
"""
predictions = tf.reduce_sum(
tf.gather(user_embeddings, sparse_ratings.indices[:, 0]) *
tf.gather(movie_embeddings, sparse_ratings.indices[:, 1]),
axis=1)
loss = tf.losses.mean_squared_error(sparse_ratings.values, predictions)
return loss
Exercise 3 (Optional): adding your own ratings to the data set¶
You have the option to add your own ratings to the data set. If you choose to do so, you will be able to see recommendations for yourself.
Start by checking the box below. Running the next cell will authenticate you to your google Drive account, and create a spreadsheet, that contains all movie titles in column 'A'. Follow the link to the spreadsheet and take 3 minutes to rate some of the movies. Your ratings should be entered in column 'B'.
USER_RATINGS = True #@param {type:"boolean"}
# @title Run to create a spreadsheet, then use it to enter your ratings.
# Authenticate user.
if USER_RATINGS:
auth.authenticate_user()
gc = gspread.authorize(GoogleCredentials.get_application_default())
# Create the spreadsheet and print a link to it.
try:
sh = gc.open('MovieLens-test')
except(gspread.SpreadsheetNotFound):
sh = gc.create('MovieLens-test')
worksheet = sh.sheet1
titles = movies['title'].values
cell_list = worksheet.range(1, 1, len(titles), 1)
for cell, title in zip(cell_list, titles):
cell.value = title
worksheet.update_cells(cell_list)
print("Link to the spreadsheet: "
"https://docs.google.com/spreadsheets/d/{}/edit".format(sh.id))
Run the next cell to load your ratings and add them to the main ratings DataFrame.
# @title Run to load your ratings.
# Load the ratings from the spreadsheet and create a DataFrame.
if USER_RATINGS:
my_ratings = pd.DataFrame.from_records(worksheet.get_all_values()).reset_index()
my_ratings = my_ratings[my_ratings[1] != '']
my_ratings = pd.DataFrame({
'user_id': "20",
'movie_id': list(map(str, my_ratings['index'])),
'rating': list(map(float, my_ratings[1])),
})
# Remove previous ratings.
ratings = ratings[ratings.user_id != "20"]
# Add new ratings.
ratings = ratings.append(my_ratings, ignore_index=True)
# Add new user to the users DataFrame.
if users.shape[0] == 20:
users = users.append(users.iloc[942], ignore_index=True)
users["user_id"][20] = "20"
print("Added your %d ratings; you have great taste!" % len(my_ratings))
ratings[ratings.user_id=="20"].merge(movies[['movie_id', 'title']])
# @title CFModel helper class (run this cell)
class CFModel(object):
"""Simple class that represents a collaborative filtering model"""
def __init__(self, embedding_vars, loss, metrics=None):
"""Initializes a CFModel.
Args:
embedding_vars: A dictionary of tf.Variables.
loss: A float Tensor. The loss to optimize.
metrics: optional list of dictionaries of Tensors. The metrics in each
dictionary will be plotted in a separate figure during training.
"""
self._embedding_vars = embedding_vars
self._loss = loss
self._metrics = metrics
self._embeddings = {k: None for k in embedding_vars}
self._session = None
@property
def embeddings(self):
"""The embeddings dictionary."""
return self._embeddings
def train(self, num_iterations=100, learning_rate=1.0, plot_results=True,
optimizer=tf.train.GradientDescentOptimizer):
"""Trains the model.
Args:
iterations: number of iterations to run.
learning_rate: optimizer learning rate.
plot_results: whether to plot the results at the end of training.
optimizer: the optimizer to use. Default to GradientDescentOptimizer.
Returns:
The metrics dictionary evaluated at the last iteration.
"""
with self._loss.graph.as_default():
opt = optimizer(learning_rate)
train_op = opt.minimize(self._loss)
local_init_op = tf.group(
tf.variables_initializer(opt.variables()),
tf.local_variables_initializer())
if self._session is None:
self._session = tf.Session()
with self._session.as_default():
self._session.run(tf.global_variables_initializer())
self._session.run(tf.tables_initializer())
tf.train.start_queue_runners()
with self._session.as_default():
local_init_op.run()
iterations = []
metrics = self._metrics or ({},)
metrics_vals = [collections.defaultdict(list) for _ in self._metrics]
# Train and append results.
for i in range(num_iterations + 1):
_, results = self._session.run((train_op, metrics))
if (i % 10 == 0) or i == num_iterations:
print("\r iteration %d: " % i + ", ".join(
["%s=%f" % (k, v) for r in results for k, v in r.items()]),
end='')
iterations.append(i)
for metric_val, result in zip(metrics_vals, results):
for k, v in result.items():
metric_val[k].append(v)
for k, v in self._embedding_vars.items():
self._embeddings[k] = v.eval()
if plot_results:
# Plot the metrics.
num_subplots = len(metrics)+1
fig = plt.figure()
fig.set_size_inches(num_subplots*10, 8)
for i, metric_vals in enumerate(metrics_vals):
ax = fig.add_subplot(1, num_subplots, i+1)
for k, v in metric_vals.items():
ax.plot(iterations, v, label=k)
ax.set_xlim([1, num_iterations])
ax.legend()
return results
Exercise 4: Build a Matrix Factorization model and train it¶
Using your sparse_mean_square_error function, write a function that builds a CFModel by creating the embedding variables and the train and test losses.
#@title Solution
def build_model(ratings, embedding_dim=3, init_stddev=1.):
"""
Args:
ratings: a DataFrame of the ratings
embedding_dim: the dimension of the embedding vectors.
init_stddev: float, the standard deviation of the random initial embeddings.
Returns:
model: a CFModel.
"""
# Split the ratings DataFrame into train and test.
train_ratings, test_ratings = split_dataframe(ratings)
# SparseTensor representation of the train and test datasets.
A_train = build_rating_sparse_tensor(train_ratings)
A_test = build_rating_sparse_tensor(test_ratings)
# Initialize the embeddings using a normal distribution.
U = tf.Variable(tf.random_normal(
[A_train.dense_shape[0], embedding_dim], stddev=init_stddev))
V = tf.Variable(tf.random_normal(
[A_train.dense_shape[1], embedding_dim], stddev=init_stddev))
train_loss = sparse_mean_square_error(A_train, U, V)
test_loss = sparse_mean_square_error(A_test, U, V)
metrics = {
'train_error': train_loss,
'test_error': test_loss
}
embeddings = {
"user_id": U,
"movie_id": V
}
return CFModel(embeddings, train_loss, [metrics])
Great, now it's time to train the model!
Go ahead and run the next cell, trying different parameters (embedding dimension, learning rate, iterations). The training and test errors are plotted at the end of training. You can inspect these values to validate the hyper-parameters.
Note: by calling model.train again, the model will continue training starting from the current values of the embeddings.
# Build the CF model and train it.
model = build_model(ratings, embedding_dim=30, init_stddev=0.5)
model.train(num_iterations=1000, learning_rate=10.)
The movie and user embeddings are also displayed in the right figure. When the embedding dimension is greater than 3, the embeddings are projected on the first 3 dimensions. The next section will have a more detailed look at the embeddings.
IV. Inspecting the Embeddings¶
In this section, we take a closer look at the learned embeddings, by
- computing your recommendations
- looking at the nearest neighbors of some movies,
- looking at the norms of the movie embeddings,
- visualizing the embedding in a projected embedding space.
#@title Solution
DOT = 'dot'
COSINE = 'cosine'
def compute_scores(query_embedding, item_embeddings, measure=DOT):
"""Computes the scores of the candidates given a query.
Args:
query_embedding: a vector of shape [k], representing the query embedding.
item_embeddings: a matrix of shape [N, k], such that row i is the embedding
of item i.
measure: a string specifying the similarity measure to be used. Can be
either DOT or COSINE.
Returns:
scores: a vector of shape [N], such that scores[i] is the score of item i.
"""
u = query_embedding
V = item_embeddings
if measure == COSINE:
V = V / np.linalg.norm(V, axis=1, keepdims=True)
u = u / np.linalg.norm(u)
scores = u.dot(V.T)
return scores
# @title User recommendations and nearest neighbors (run this cell)
def user_recommendations(model, measure=DOT, exclude_rated=False, k=6):
if USER_RATINGS:
scores = compute_scores(
model.embeddings["user_id"][20], model.embeddings["movie_id"], measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'movie_id': movies['movie_id'],
'titles': movies['title'],
'genres': movies['all_genres'],
})
if exclude_rated:
# remove movies that are already rated
rated_movies = ratings[ratings.user_id == "20"]["movie_id"].values
df = df[df.movie_id.apply(lambda movie_id: movie_id not in rated_movies)]
display.display(df.sort_values([score_key], ascending=False).head(k))
def movie_neighbors(model, title_substring, measure=DOT, k=10):
# Search for movie ids that match the given substring.
ids = movies[movies['title'].str.contains(title_substring)].index.values
titles = movies.iloc[ids]['title'].values
if len(titles) == 0:
raise ValueError("Found no movies with title %s" % title_substring)
print("Nearest neighbors of : %s." % titles[0])
if len(titles) > 1:
print("[Found more than one matching movie. Other candidates: {}]".format(
", ".join(titles[1:])))
movie_id = ids[0]
scores = compute_scores(
model.embeddings["movie_id"][movie_id], model.embeddings["movie_id"],
measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'titles': movies['title'],
'genres': movies['all_genres']
})
display.display(df.sort_values([score_key], ascending=False).head(k))
# @title User recommendations and nearest neighbors (run this cell)# mark for deletion
def user_recommendations(model, measure=DOT, exclude_rated=False, k=6):
if USER_RATINGS:
scores = compute_scores(
model.embeddings["user_id"][20], model.embeddings["movie_id"], measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'movie_id': movies['movie_id'],
'titles': movies['title'],
'genres': movies['all_genres'],
})
if exclude_rated:
# remove movies that are already rated
rated_movies = ratings[ratings.user_id == "20"]["movie_id"].values
df = df[df.movie_id.apply(lambda movie_id: movie_id not in rated_movies)]
display.display(df.sort_values([score_key], ascending=False).head(k))
def movie_neighbors(model, title_substring, measure=DOT, k=12):
# Search for movie ids that match the given substring.
ids = movies[movies['title'].str.contains(title_substring)].index.values
titles = movies.iloc[ids]['title'].values
if len(titles) == 0:
raise ValueError("Found no movies with title %s" % title_substring)
print("Nearest neighbors of : %s." % titles[0])
if len(titles) > 1:
print("[Found more than one matching movie. Other candidates: {}]".format(
", ".join(titles[1:])))
movie_id = ids[0]
scores = compute_scores(
model.embeddings["movie_id"][movie_id], model.embeddings["movie_id"],
measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'titles': movies['title'],
'genres': movies['all_genres']
})
display.display(df.sort_values([score_key], ascending=False).head(k))
Equipped with this function, we can compute recommendations, where the query embedding can be either a user embedding or a movie embedding.
Your recommendations¶
If you chose to input your recommendations, you can run the next cell to generate recommendations for you.
user_recommendations(model, measure=COSINE, k=10, exclude_rated=False)
How do the recommendations look?
Movie Nearest neighbors¶
Let's look at the neareast neighbors for some of the movies.
movie_neighbors(model, "Aladdin", DOT)
movie_neighbors(model, "Aladdin", COSINE)
It seems that the quality of learned embeddings may not be very good. This will be addressed in Section V by adding several regularization techniques. First, we will further inspect the embeddings.
Movie Embedding Norm¶
We can also observe that the recommendations with dot-product and cosine are different: with dot-product, the model tends to recommend popular movies. This can be explained by the fact that in matrix factorization models, the norm of the embedding is often correlated with popularity (popular movies have a larger norm), which makes it more likely to recommend more popular items. We can confirm this hypothesis by sorting the movies by their embedding norm, as done in the next cell.
# @title Embedding Visualization code (run this cell)
def movie_embedding_norm(models):
"""Visualizes the norm and number of ratings of the movie embeddings.
Args:
model: A MFModel object.
"""
if not isinstance(models, list):
models = [models]
df = pd.DataFrame({
'title': movies['title'],
'genre': movies['genre'],
'num_ratings': movies_ratings['rating count'],
})
charts = []
brush = alt.selection_interval()
for i, model in enumerate(models):
norm_key = 'norm'+str(i)
df[norm_key] = np.linalg.norm(model.embeddings["movie_id"], axis=1)
nearest = alt.selection(
type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
empty='none')
base = alt.Chart().mark_circle().encode(
x='num_ratings',
y=norm_key,
color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
).properties(
selection=nearest).add_selection(brush)
text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
x='num_ratings', y=norm_key,
text=alt.condition(nearest, 'title', alt.value('')))
charts.append(alt.layer(base, text))
return alt.hconcat(*charts, data=df)
def visualize_movie_embeddings(data, x, y):
nearest = alt.selection(
type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
empty='none')
base = alt.Chart().mark_circle().encode(
x=x,
y=y,
color=alt.condition(genre_filter, "genre", alt.value("whitesmoke")),
).properties(
width=600,
height=600,
selection=nearest)
text = alt.Chart().mark_text(align='left', dx=5, dy=-5).encode(
x=x,
y=y,
text=alt.condition(nearest, 'title', alt.value('')))
return alt.hconcat(alt.layer(base, text), genre_chart, data=data)
def tsne_movie_embeddings(model):
"""Visualizes the movie embeddings, projected using t-SNE with Cosine measure.
Args:
model: A MFModel object.
"""
tsne = sklearn.manifold.TSNE(
n_components=2, perplexity=40, metric='cosine', early_exaggeration=10.0,
init='pca', verbose=True, n_iter=400)
print('Running t-SNE...')
V_proj = tsne.fit_transform(model.embeddings["movie_id"])
movies.loc[:,'x'] = V_proj[:, 0]
movies.loc[:,'y'] = V_proj[:, 1]
return visualize_movie_embeddings(movies, 'x', 'y')
tsne_movie_embeddings(model)
movie_embedding_norm(model)
Note: Depending on how the model is initialized, you may observe that some niche movies (ones with few ratings) have a high norm, leading to spurious recommendations. This can happen if the embedding of that movie happens to be initialized with a high norm. Then, because the movie has few ratings, it is infrequently updated, and can keep its high norm. This will be alleviated by using regularization.
Try changing the value of the hyper-parameter init_stddev. One quantity that can be helpful is that the expected norm of a $d$-dimensional vector with entries $\sim \mathcal N(0, \sigma^2)$ is approximatley $\sigma \sqrt d$.
How does this affect the embedding norm distribution, and the ranking of the top-norm movies?
#@title Solution
model_lowinit = build_model(ratings, embedding_dim=30, init_stddev=0.05)
model_lowinit.train(num_iterations=1000, learning_rate=10.)
movie_neighbors(model_lowinit, "Aladdin", DOT)
movie_neighbors(model_lowinit, "Aladdin", COSINE)
movie_embedding_norm([model, model_lowinit])
Embedding visualization¶
Since it is hard to visualize embeddings in a higher-dimensional space (when the embedding dimension $k > 3$), one approach is to project the embeddings to a lower dimensional space. T-SNE (T-distributed Stochastic Neighbor Embedding) is an algorithm that projects the embeddings while attempting to preserve their pariwise distances. It can be useful for visualization, but one should use it with care. For more information on using t-SNE, see How to Use t-SNE Effectively.
tsne_movie_embeddings(model_lowinit)
You can highlight the embeddings of a given genre by clicking on the genres panel (SHIFT+click to select multiple genres).
We can observe that the embeddings do not seem to have any notable structure, and the embeddings of a given genre are located all over the embedding space. This confirms the poor quality of the learned embeddings. One of the main reasons, which we will address in the next section, is that we only trained the model on observed pairs, and without regularization.
V. Regularization In Matrix Factorization¶
# @title Solution
def gravity(U, V):
"""Creates a gravity loss given two embedding matrices."""
return 1. / (U.shape[0].value*V.shape[0].value) * tf.reduce_sum(
tf.matmul(U, U, transpose_a=True) * tf.matmul(V, V, transpose_a=True))
def build_regularized_model(
ratings, embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
init_stddev=0.1):
"""
Args:
ratings: the DataFrame of movie ratings.
embedding_dim: The dimension of the embedding space.
regularization_coeff: The regularization coefficient lambda.
gravity_coeff: The gravity regularization coefficient lambda_g.
Returns:
A CFModel object that uses a regularized loss.
"""
# Split the ratings DataFrame into train and test.
train_ratings, test_ratings = split_dataframe(ratings)
# SparseTensor representation of the train and test datasets.
A_train = build_rating_sparse_tensor(train_ratings)
A_test = build_rating_sparse_tensor(test_ratings)
U = tf.Variable(tf.random_normal(
[A_train.dense_shape[0], embedding_dim], stddev=init_stddev))
V = tf.Variable(tf.random_normal(
[A_train.dense_shape[1], embedding_dim], stddev=init_stddev))
error_train = sparse_mean_square_error(A_train, U, V)
error_test = sparse_mean_square_error(A_test, U, V)
gravity_loss = gravity_coeff * gravity(U, V)
regularization_loss = regularization_coeff * (
tf.reduce_sum(U*U)/U.shape[0].value + tf.reduce_sum(V*V)/V.shape[0].value)
total_loss = error_train + regularization_loss + gravity_loss
losses = {
'train_error_observed': error_train,
'test_error_observed': error_test,
}
loss_components = {
'observed_loss': error_train,
'regularization_loss': regularization_loss,
'gravity_loss': gravity_loss,
}
embeddings = {"user_id": U, "movie_id": V}
return CFModel(embeddings, total_loss, [losses, loss_components])
It is now time to train the regularized model! You can try different values of the regularization coefficients, and different embedding dimensions.
reg_model = build_regularized_model(
ratings, regularization_coeff=0.1, gravity_coeff=1.0, embedding_dim=35,
init_stddev=.05)
reg_model.train(num_iterations=2000, learning_rate=20.)
Observe that adding the regularization terms results in a higher MSE, both on the training and test set. However, as we will see, the quality of the recommendations improves. This highlights a tension between fitting the observed data and minimizing the regularization terms. Fitting the observed data often emphasizes learning high similarity (between items with many interactions), but a good embedding representation also requires learning low similarity (between items with few or no interactions).
Inspect the results¶
Let's see if the results with regularization look better.
user_recommendations(reg_model, DOT, exclude_rated=True, k=10)
movie_neighbors(reg_model, "Aladdin", DOT)
movie_neighbors(reg_model, "Aladdin", COSINE)
Here we compare the embedding norms for model and reg_model. Selecting a subset of the embeddings will highlight them on both charts simultaneously.
movie_embedding_norm([model, model_lowinit, reg_model])
# Visualize the embeddings
tsne_movie_embeddings(reg_model)
We should observe that the embeddings have a lot more structure than the unregularized case. Try selecting different genres and observe how they tend to form clusters (for example Horror, Animation and Children).
Conclusion¶
This concludes this section on matrix factorization models. Note that while the scale of the problem is small enough to allow efficient training using SGD, many practical problems need to be trained using more specialized algorithms such as Alternating Least Squares (see tf.contrib.factorization.WALSMatrixFactorization for a TF implementation).
VI. Softmax model¶
In this section, we will train a simple softmax model that predicts whether a given user has rated a movie.
Note: if you are taking the self-study version of the class, make sure to read through the part of the class covering Softmax training before working on this part.
The model will take as input a feature vector $x$ representing the list of movies the user has rated. We start from the ratings DataFrame, which we group by user_id.
rated_movies = (ratings[["user_id", "movie_id"]]
.groupby("user_id", as_index=False)
.aggregate(lambda x: list(x)))
rated_movies
rated_movies[rated_movies['user_id'] == "20"]
We then create a function that generates an example batch, such that each example contains the following features:
- movie_id: A tensor of strings of the movie ids that the user rated.
- genre: A tensor of strings of the genres of those movies
- year: A tensor of strings of the release year.
#@title Batch generation code (run this cell)
years_dict = {
movie: year for movie, year in zip(movies["movie_id"], movies["year"])
}
genres_dict = {
movie: genres.split('-')
for movie, genres in zip(movies["movie_id"], movies["all_genres"])
}
def make_batch(ratings, batch_size):
"""Creates a batch of examples.
Args:
ratings: A DataFrame of ratings such that examples["movie_id"] is a list of
movies rated by a user.
batch_size: The batch size.
"""
def pad(x, fill):
return pd.DataFrame.from_dict(x).fillna(fill).values
movie = []
year = []
genre = []
label = []
for movie_ids in ratings["movie_id"].values:
movie.append(movie_ids)
genre.append([x for movie_id in movie_ids for x in genres_dict[movie_id]])
year.append([years_dict[movie_id] for movie_id in movie_ids])
label.append([int(movie_id) for movie_id in movie_ids])
features = {
"movie_id": pad(movie, ""),
"year": pad(year, ""),
"genre": pad(genre, ""),
"label": pad(label, -1)
}
batch = (
tf.data.Dataset.from_tensor_slices(features)
.shuffle(1000)
.repeat()
.batch(batch_size)
.make_one_shot_iterator()
.get_next())
return batch
def select_random(x):
"""Selectes a random elements from each row of x."""
def to_float(x):
return tf.cast(x, tf.float32)
def to_int(x):
return tf.cast(x, tf.int64)
batch_size = tf.shape(x)[0]
rn = tf.range(batch_size)
nnz = to_float(tf.count_nonzero(x >= 0, axis=1))
rnd = tf.random_uniform([batch_size])
ids = tf.stack([to_int(rn), to_int(nnz * rnd)], axis=1)
return to_int(tf.gather_nd(x, ids))
Exercise 7: Write a loss function for the softmax model.¶
In this exercise, we will write a function that takes tensors representing the user embeddings $\psi(x)$, movie embeddings $V$, target label $y$, and return the cross-entropy loss.
Hint: You can use the function tf.nn.sparse_softmax_cross_entropy_with_logits, which takes logits as input, where logits refers to the product $\psi(x) V^\top$.
# @title Solution
def softmax_loss(user_embeddings, movie_embeddings, labels):
"""Returns the cross-entropy loss of the softmax model.
Args:
user_embeddings: A tensor of shape [batch_size, embedding_dim].
movie_embeddings: A tensor of shape [num_movies, embedding_dim].
labels: A tensor of [batch_size], such that labels[i] is the target label
for example i.
Returns:
The mean cross-entropy loss.
"""
# Verify that the embddings have compatible dimensions
user_emb_dim = user_embeddings.shape[1].value
movie_emb_dim = movie_embeddings.shape[1].value
if user_emb_dim != movie_emb_dim:
raise ValueError(
"The user embedding dimension %d should match the movie embedding "
"dimension % d" % (user_emb_dim, movie_emb_dim))
logits = tf.matmul(user_embeddings, movie_embeddings, transpose_b=True)
loss = tf.reduce_mean(tf.nn.sparse_softmax_cross_entropy_with_logits(
logits=logits, labels=labels))
return loss
Exercise 8: Build a softmax model, train it, and inspect its embeddings.¶
We are now ready to build a softmax CFModel. Complete the build_softmax_model function in the next cell. The architecture of the model is defined in the function create_user_embeddings and illustrated in the figure below. The input embeddings (movie_id, genre and year) are concatenated to form the input layer, then we have hidden layers with dimensions specified by the hidden_dims argument. Finally, the last hidden layer is multiplied by the movie embeddings to obtain the logits layer. For the target label, we will use a randomly-sampled movie_id from the list of movies the user rated.
Complete the function below by creating the feature columns and embedding columns, then creating the loss tensors both for the train and test sets (using the softmax_loss function of the previous exercise).
def movie_neighbors2(model, title_substring, measure=DOT, k=12):
# Search for movie ids that match the given substring.
ids = movies[movies['title'].str.contains(title_substring)].index.values
titles = movies.iloc[ids]['title'].values
if len(titles) == 0:
raise ValueError("Found no movies with title %s" % title_substring)
print("Nearest neighbors of : %s." % titles[0])
if len(titles) > 1:
print("[Found more than one matching movie. Other candidates: {}]".format(
", ".join(titles[1:])))
movie_id = ids[0]
scores = compute_scores(
model.embeddings["movie_id"][movie_id], model.embeddings["movie_id"],
measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'titles': movies['title'],
'genres': movies['all_genres']
})
display.display(df.sort_values([score_key], ascending=False).head(k))
# @title User recommendations and nearest neighbors (run this cell)
def user_recommendations2(model, measure=DOT, exclude_rated=False, k=6):
if USER_RATINGS:
scores = compute_scores(
model.embeddings["user_id"][20], model.embeddings["movie_id"], measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'movie_id': movies['movie_id'],
'titles': movies['title'],
'genres': movies['all_genres'],
})
if exclude_rated:
# remove movies that are already rated
rated_movies = ratings[ratings.user_id == "20"]["movie_id"].values
df = df[df.movie_id.apply(lambda movie_id: movie_id not in rated_movies)]
display.display(df.sort_values([score_key], ascending=False).head(k))
# @title User recommendations and nearest neighbors (run this cell)
def user_recommendations3(model, measure=DOT, exclude_rated=False, k=6):
if USER_RATINGS:
scores = compute_scores(
model.embeddings["user_idtest"][20], model.embeddings["movie_id"], measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'movie_id': movies['movie_id'],
'titles': movies['title'],
'genres': movies['all_genres'],
})
if exclude_rated:
# remove movies that are already rated
rated_movies = ratings[ratings.user_id == "20"]["movie_id"].values
df = df[df.movie_id.apply(lambda movie_id: movie_id not in rated_movies)]
display.display(df.sort_values([score_key], ascending=False).head(k))
user_recommendations3(softmax_model, COSINE, exclude_rated=True, k=10)
user_recommendations3(softmax_model, DOT, exclude_rated=True, k=10)
user_recommendations3(softmax_model, DOT, exclude_rated=False, k=10)
user_recommendations2(reg_model, DOT, exclude_rated=True, k=10)
user_recommendations2(softmax_model, DOT, exclude_rated=True, k=10)
user_recommendations2(softmax_model, DOT, exclude_rated=True, k=20)
# @title Solution
def build_softmax_model(rated_movies, embedding_cols, hidden_dims):
"""Builds a Softmax model for MovieLens.
Args:
rated_movies: DataFrame of traing examples.
embedding_cols: A dictionary mapping feature names (string) to embedding
column objects. This will be used in tf.feature_column.input_layer() to
create the input layer.
hidden_dims: int list of the dimensions of the hidden layers.
Returns:
A CFModel object.
"""
def create_network(features):
"""Maps input features dictionary to user embeddings.
Args:
features: A dictionary of input string tensors.
Returns:
outputs: A tensor of shape [batch_size, embedding_dim].
"""
# Create a bag-of-words embedding for each sparse feature.
inputs = tf.feature_column.input_layer(features, embedding_cols)
# Hidden layers.
input_dim = inputs.shape[1].value
for i, output_dim in enumerate(hidden_dims):
w = tf.get_variable(
"hidden%d_w_" % i, shape=[input_dim, output_dim],
initializer=tf.truncated_normal_initializer(
stddev=1./np.sqrt(output_dim))) / 10.
outputs = tf.matmul(inputs, w)
input_dim = output_dim
inputs = outputs
return outputs
train_rated_movies, test_rated_movies = split_dataframe(rated_movies)
train_batch = make_batch(train_rated_movies, 800)
test_batch = make_batch(test_rated_movies, 800)
with tf.variable_scope("model", reuse=False):
# Train
train_user_embeddings = create_network(train_batch)
train_labels = select_random(train_batch["label"])
with tf.variable_scope("model", reuse=True):
# Test
test_user_embeddings = create_network(test_batch)
test_labels = select_random(test_batch["label"])
movie_embeddings = tf.get_variable(
"input_layer/movie_id_embedding/embedding_weights")
test_loss = softmax_loss(
test_user_embeddings, movie_embeddings, test_labels)
train_loss = softmax_loss(
train_user_embeddings, movie_embeddings, train_labels)
_, test_precision_at_10 = tf.metrics.precision_at_k(
labels=test_labels,
predictions=tf.matmul(test_user_embeddings, movie_embeddings, transpose_b=True),
k=10)
metrics = (
{"train_loss": train_loss, "test_loss": test_loss},
{"test_precision_at_10": test_precision_at_10}
)
embeddings = {"movie_id": movie_embeddings, "user_id": train_user_embeddings, "user_idtest": test_user_embeddings}
return CFModel(embeddings, train_loss, metrics)
Train the Softmax model¶
We are now ready to train the softmax model. You can set the following hyperparameters:
- learning rate
- number of iterations. Note: you can run
softmax_model.train()again to continue training the model from its current state. - input embedding dimensions (the
input_dimsargument) - number of hidden layers and size of each layer (the
hidden_dimsargument)
Note: since our input features are string-valued (movie_id, genre, and year), we need to map them to integer ids. This is done using tf.feature_column.categorical_column_with_vocabulary_list, which takes a vocabulary list specifying all the values the feature can take. Then each id is mapped to an embedding vector using tf.feature_column.embedding_column.
# Create feature embedding columns
def make_embedding_col(key, embedding_dim):
categorical_col = tf.feature_column.categorical_column_with_vocabulary_list(
key=key, vocabulary_list=list(set(movies[key].values)), num_oov_buckets=0)
return tf.feature_column.embedding_column(
categorical_column=categorical_col, dimension=embedding_dim,
# default initializer: trancated normal with stddev=1/sqrt(dimension)
combiner='mean')
with tf.Graph().as_default():
softmax_model = build_softmax_model(
rated_movies,
embedding_cols=[
make_embedding_col("movie_id", 35),
make_embedding_col("genre", 3),
make_embedding_col("year", 2),
],
hidden_dims=[35])
softmax_model.train(
learning_rate=8., num_iterations=3000, optimizer=tf.train.AdagradOptimizer)
Inspect the embeddings¶
We can inspect the movie embeddings as we did for the previous models. Note that in this case, the movie embeddings are used at the same time as input embeddings (for the bag of words representation of the user history), and as softmax weights.
user_recommendations2(reg_model, DOT, exclude_rated=True, k=10)
user_recommendations2(softmax_model, DOT, exclude_rated=True, k=10)
user_recommendations2(reg_model, DOT, exclude_rated=True, k=10)
movie_neighbors(softmax_model, "Aladdin", DOT)
movie_neighbors(softmax_model, "Aladdin", COSINE)
movie_embedding_norm([reg_model, softmax_model])
tsne_movie_embeddings(softmax_model)
tsne_movie_embeddings(reg_model)
user_recommendations(softmax_model, measure=COSINE, k=10)
Congratulations!¶
You have completed this Colab notebook.
If you would like to further explore these models, we encourage you to try different hyperparameters and observe how this affects the quality of the model and the structure of the embedding space. Here are some suggestions:
- Change the embedding dimension.
- In the softmax model: change the number of hidden layers, and the input features. For example, you can try a model with no hidden layers, and only the movie ids as inputs.
- Using other similarity measures: In this Colab notebook, we used dot product $d(u, V_j) = \langle u, V_j \rangle$ and cosine $d(u, V_j) = \frac{\langle u, V_j \rangle}{\|u\|\|V_j\|}$, and discussed how the norms of the embeddings affect the recommendations. You can also try other variants which apply a transformation to the norm, for example $d(u, V_j) = \frac{\langle u, V_j \rangle}{\|V_j\|^\alpha}$.